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We extend the well-known Sacks-Uhlenbeck energy gap result (1981) forharmonic maps from closed Riemann surfaces into closed Riemannian manifoldsfrom the case of maps with small energy (thus near a constant map), to the caseof harmonic maps with high absolute energy but small energy relative to areference harmonic map.

Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds